Hoon Hong. Quantifier elimination for formulas constrained by quadratic equations via slope resultants. THE Computer Journal, 36(5):440--449, 1993. Special issue on computational quantifier elimination.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....for various special cases of the general quantifier elimination problem for the first order theory of real numbers. These include the elimination of one existential quantifier 9x in front of quantifier free formulas restricted by a non trivial quadratic equation in x (the case considered also in [7]) and more generally in front of arbitrary quantifier free formulas involving only polynomials that are quadratic in x. The method generalizes the linear quantifier elimination method by virtual substitution of test terms in [9] It yields a quantifier elimination method for an arbitrary number ....

....procedures for subproblems given by input formulas of special forms. The case of input formulas in which all quantified variables occur only linearly has been handled in [13, 9] Elimination of a single quantified variable that is restricted by a non trivial quadratic equation has been treated in [7]. In both cases the specialized elementary methods perform significantly better than the general purpose method in [3, 6] in some test examples. Notice, however, that the theoretical worst case complexity of the quantifier elimination problem for the linear case is the same as for the general ....

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H. Hong, Quantifier elimination for formulas constrained by quadratic equation, The Computer Journal 36, 5 (1993), pp. 440-449.

....rank. For pure Toeplitz matrices, for example, we have ff 2. Toeplitz like matrices are ubiquitous in symbolic computation as resultants and subresultants have this form (Brown and Traub 1971, Sasaki and 226 PASCO 94: First International Symposium on Parallel Symbolic Computation Furukawa 1984, Hong 1993). Furthermore, block matrices are used, for instance, as a parallelization technique by reducing the dimensions while increasing the running time of the arithmetic operations on the individual entries, which now are matrices rather than field elements. This approach leads to coarse grain parallel ....

Hong, H., "Quantifier elimination for formulas constrained by quadratic equations via slope resultants," The Computer J. 36/5, pp. 439-- 449 (1993).

....[DH88, Wei88] Thus the attention turned to special procedures for restricted problem classes, where the elimination procedures can be tuned to the structure of the problem. The focus was on considering formulas in which the occurrence of quantified variables is restricted to low degrees [Hon93b, Hon93c, GV93] This was initiated by a theoretical paper of Weispfenning in 1988 [Wei88] who introduced the method of virtual disjunctive term substitution, which has been continuously extended and improved [LW93, Wei94, Wei97a, DSW96, DS97b] The worst case complexity of this method is doubly ....

Hoon Hong. Quantifier elimination for formulas constrained by quadratic equations via slope resultants. THE Computer Journal, 36(5):440--449, 1993. Special issue on computational quantifier elimination.

....classes [DH88, Wei88] Thus the attention turned to special procedures for restricted problem classes, where the elimination procedures can be tuned to the structure of the problem. The focus was on considering formulas in which the occurrence of quantified variables is restricted to low degrees [Hon93b, Hon93c, GV93] This was initiated by a theoretical paper of Weispfenning in 1988 [Wei88] who introduced the method of virtual disjunctive term substitution, which has been continuously extended and improved [LW93, Wei94, Wei97a, DSW96, DS97b] The worst case complexity of this method is doubly ....

Hoon Hong. Quantifier elimination for formulas constrained by quadratic equations. In Manuel Bronstein, editor, Proceedings of the 1993 International Symposium on Symbolic and Algebraic Computation (ISSAC 93), pages 264--274, Kiev, Ukraine, July 1993. ACM, ACM Press.

....Thus the attention turned to special procedures for restricted problem classes, where the elimination procedures can be tuned to the structure of the problem. The focus was on considering formulas in which the occurrence of quantified variables is restricted to low degrees, cf. Wei88,LW93,Hon93a,Hon93b,GV93,Wei94b,Wei97a] This was initiated by the third author in 1988. In his virtual substitution method the number of parameters plays a minor role for the complexity. The worst case complexity of the method is doubly exponential only in the number of the quantifier blocks of the input formula. ....

....remainder of need be considered. This idea can easily be extended to a quadratic equation instead of a linear one, taking into account again the discriminant. Ideas very similar to our extended Gauss elimination have independently been considered by Hong within the cad framework, cf. Hon93b] An extended quantifier elimination can be straightforwardly derived from this method by not constructing a disjunction at the end. Instead all the quantifierfree substitution results are kept separately together with the candidate terms yielding them. The notion of virtual substitution refers ....

Hoon Hong. Quantifier elimination for formulas constrained by quadratic equations via slope resultants. THE Computer Journal, 36(5):440--449, 1993. Special issue on computational quantifier elimination.

....Thus the attention turned to special procedures for restricted problem classes, where the elimination procedures can be tuned to the structure of the problem. The focus was on considering formulas in which the occurrence of quantified variables is restricted to low degrees, cf. Wei88,LW93,Hon93a,Hon93b,GV93,Wei94b,Wei97a] This was initiated by the third author in 1988. In his virtual substitution method the number of parameters plays a minor role for the complexity. The worst case complexity of the method is doubly exponential only in the number of the quantifier blocks of the input ....

Hoon Hong. Quantifier elimination for formulas constrained by quadratic equations. In Manuel Bronstein, editor, Proceedings of the International Symposium on Symbolic and Algebraic Computation (ISSAC 93), pages 264-- 274, New York, July 1993. ACM, ACM Press.

....method will be of practical significance. 1 Introduction The great theoretical and practical complexity of the quantifier elimination problem for the elementary theory of the reals (compare [Renegar, Weispfenning 1] has stimulated research in special purpose quantifier elimination methods (see [Hong 2, Weispfenning 1, Loos Weispf. Weispfenning 2, Weispfenning 3, Gonzales Vega] for input formulas of special types. By exploiting the special features of the input formulas considered, these procedures tend to perform significantly better that the general purpose CAD based algorithm of ....

....2, Weispfenning 3, Gonzales Vega] for input formulas of special types. By exploiting the special features of the input formulas considered, these procedures tend to perform significantly better that the general purpose CAD based algorithm of CollinsHong ( Collins Hong, Hong 1] compare [Hong 2, Weispfenning 2] The present note is another contribution to this research program. It concerns input formulas of the form 9x( where is an arbitrary boolean combination of polynomial equations and inequalities in x and other (free) variables y 1 ; yn with rational coefficients ....

[Article contains additional citation context not shown here]

H. Hong, Quantifier elimination for formulas constrained by quadratic equation, The Computer Journal 36, 5 (1993), pp. 440-449.

....is asymptotically worst case optimal up to a constant. It achieves a worst case computing time bound which is polynomial in the length of the input formula, exponential in the number of quantified variables, and doubly exponential in the number of quantifier blocks only. Until recently (compare [9]) research on the implementation of quantifier elimination algorithms has been concentrated on the full elementary theory of real closed fields; there is a widespread belief [5] starting with the original work of Tarski [13] that these algorithms are of a rather academic nature and are not ....

Hong H. (1993) Quantifier Elimination for Formulas Constrained by Quadratic Equations, Proc. ISSAC'93, to appear.

....engineering can be reduced to the problem of testing positiveness of polynomials. In 1930, Tarski [33, 34] showed that the problem is decidable. In fact, he gave a decision method for a more general problem than just testing positiveness. Since then, many improvements and new methods were proposed [7, 1, 26, 2, 3, 27, 13, 35, 30, 14, 15, 16, 9, 29, 17, 28, 6, 20, 19, 18, 25, 36, 8, 21, 22, 5, 12, 23]. However, these methods are computationally expensive due to their generality. Naturally one is interested in devising more efficient methods for the sub problem: testing positiveness. But then, this sub problem turns out to be still difficult. Thus, several authors (mainly from the field of term ....

H. Hong. Quantifier elimination for formulas constrained by quadratic equations via slope resultants. The Computer Journal, 36(5):440--449, 1993.

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Hoon Hong. Quantifier elimination for formulas constrained by quadratic equations via slope resultants. THE Computer Journal, 36(5):440--449, 1993. Special issue on computational quantifier elimination.

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H. Hong, Quantifier elimination for formulas constrained by quadratic equation, The Computer Journal 36, 5 (1993), pp. 440-449.

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H. Hong, Quantifier elimination for formulas constrained by quadratic equation, The Computer Journal 36, 5 (1993), pp. 440-449.

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